+0  
 
0
118
1
avatar

What is the constant term in the expansion of \(\left(\sqrt{x}+\dfrac5x\right)^{9}\)?

 Jul 11, 2020
 #1
avatar+8341 
0

Considering the general term of the expansion:

 

\(\left(\sqrt x + \dfrac5x\right)^9 = \displaystyle\sum_{k = 0}^9 \binom{9}k\cdot \left(\sqrt x\right)^k \left(\dfrac5x\right)^{9-k}\)

 

Simplifying,

 

\(\left(\sqrt x + \dfrac5x\right)^9 = \displaystyle\sum_{k = 0}^9 \binom{9}k \cdot 5^{9-k}\cdot x^{3k/2-9}\)

 

When the term is constant, power of x is 0.

 

\(\dfrac{3k}2 - 9 = 0\\ k = 6\)

 

Now, substitute k = 6 into the general term, you get the constant term.

 

(Note that if k was not an integer, the constant term would be 0.)

 Jul 11, 2020

43 Online Users

avatar
avatar